Introduction to decoupling
Decoupling theory is an approach to estimating exponential sums developed by Jean Bourgain and Ciprian Demeter. One striking application is a nearly sharp estimate in the Vinogradov mean value theorem (proven jointly with me). I will explain what decoupling theorems say and then discuss some of the main ideas in their proofs. We will focus on the simplest deocoupling theorem - decoupling for the parabola - but the ideas we describe are relevant to more complicated decoupling theorems as well. Looking at the problem at many different scales is one of the key ideas in the proofs, and we will try to explain how this works and why it helps.